Problem: Let $y=\tan(x)$. What is the value of $\dfrac{dy}{dx}$ at $x=\dfrac{3\pi}{4}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac12$ (Choice B) B $2$ (Choice C) C $-\dfrac{1}{2}$ (Choice D) D $-2$
Solution: Let's first find $\dfrac{dy}{dx}$. Then, we can evaluate it at $x=\dfrac{3\pi}{4}$. Recall that the derivative of $\tan(x)$ is $\dfrac{1}{\cos^2(x)}$, or $\sec^2(x)$. Put another way, $\dfrac{d}{dx}[\tan(x)]=\dfrac{1}{\cos^2(x)}=\sec^2(x)$. [Is there a way to know this without memorizing?] Now let's plug in $x={\dfrac{3\pi}{4}}$ : $\begin{aligned} &\phantom{=}\dfrac{1}{\cos^2\left({\dfrac{3\pi}{4}}\right)} \\\\ &=\dfrac{1}{\left(-\dfrac{\sqrt{2}}{2}\right)^2} \\\\ &=\dfrac{1}{\left(\dfrac{2}{4}\right)} \\\\ &=2 \end{aligned}$ In conclusion, the value of $\dfrac{dy}{dx}$ at $x=\dfrac{3\pi}{4}$ is $2$.